Towards sharp Bohnenblust--Hille constants

TL;DR

This paper investigates the optimal constants in Bohnenblust--Hille inequalities, providing evidence for their universal boundedness across all dimensions and introducing entropy and complexity measures to analyze their behavior.

Contribution

It introduces the notions of entropy and complexity to study the Bohnenblust--Hille constants and supports the conjecture that these constants are universally bounded regardless of the multilinear form dimension.

Findings

Entropy growth is linked to the boundedness of constants.

Optimal constants are exactly $2^{1-1/m}$ under certain entropy growth assumptions.

Classified all extremal cases for bilinear and Littlewood inequalities.

Abstract

We investigate the optimality problem associated with the best constants in a class of Bohnenblust--Hille type inequalities for $m$--linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust--Hille inequality are universally bounded, irrespectively of the value of $m$; hereafter referred as the \textit{Universality Conjecture}. In our approach, we introduce the {notions of entropy and complexity}, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to $m$, then the optimal constants of the $m$% --linear Bohnenblust--Hille inequality for real scalars are indeed bounded universally in $m$. It is likely that indeed the entropy grows as $4^{m-1}$, and in this scenario, we show that the optimal constants are precisely $2^{1-\frac{1}{m}} $. In the bilinear case, $m=2$, we show that any extremum of the Littlewood's $4/3$-inequality has entropy $4$ and complexity $2$, and thus we are able to classify all extrema of the problem. We also prove that, for any mixed $\left( \ell _{1},\ell _{2}\right) $% --Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely $(\sqrt{2})^{m-1}$. In addition to the {notions of entropy and complexity}, the approach we develop in this work makes decisive use of a family of strongly non-symmetric $m$--linear forms, which has further consequences to the theory, as we explain herein.